Wave Propagation in Linear Media | re-examined

7.3 Implementation of OscInt and related functions
the complete list of solutions. We then regard only the real part of the possibly conjugate
complex solutions and return the largest one, provided it is still larger than the lower limit a.
Thus we haveo= maxfxo;ag.
extrema = NSolve[zerof'[x] == 0,x,WorkingPrecision->wp];
If[Length[extrema] == 0 || extrema === ffgg, a,
Max[Re[x/.Last[extrema]],a]]
Remark (Extrema) To nd the extrema of (x), we cannot use FindRoot since we do
not know where to start the iteration. Hence we cannot ensure that we obtain the largest
solution. This is why wehave to use the slower NSolve. From the code it is also clear that
the function zerof must be di erentiable. The odd-looking test for extrema === ffgg
is nothing but the condition 0 (x) 0, because this structure is what NSolve returns to
indicate that the equation is satis ed for all x.
Example 7.3.1 Consider the argument function
(x) =x(x,1+j)(x,1,j)(x,4+j)(x,4,j)
=x , x 4 ,10x 3 +35x 2 ,50x +34 ;
which has no extremum at all, as the plot shows. Consequently, the rst derivative has two pairs of
conjugate complex zeros, and PartitionOffs indeed returns the larger real part of the two.
In[1]:= f[x_] := x (x-1+I) (x-1-I) (x-4+I) (x-4-I)
In[2]:= Plot[f[x],fx,-0.5,4.5g];
60
40
20
-20
1 2 3 4
In[3]:= NSolve[f'[x]==0,x]
Out[3]= ffx -> 0.740373 - 0.294381 Ig,fx -> 0.740373 + 0.294381 Ig,
fx -> 3.25963 - 0.294381 Ig,fx -> 3.25963 + 0.294381 Igg
In[4]:= PartitionOffs[f,0]
Out[4]= 3.25963
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